Almost-crystallographic groups as quotients of Artin braid groups

Let n, k >= 3. In this paper, we analyse the quotient group B-n/Gamma(k)(P-n) of the Artin braid group B-n by the subgroup Gamma(k) (P-n) belonging to the lower central series of the Artin pure braid group P-n. We prove that it is an almost-crystallographic group. We then focus more specifically on the case k = 3. If n >= 5, and if T is an element of N is such that gcd (T, 6) = 1, we show that B-n/Gamma(3)(P-n) possesses an element of order T if and only if S-n does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order T in B-n/Gamma(3) (P-n) with those of elements of order Tin the symmetric group S-n. We also exhibit a presentation for the almost-crystallographic group B-n/Gamma(3)(P-n). Finally, we obtain some 4-dimensional almost-Bieberbach subgroups of B-n/Gamma(3) (P-3), we explain how to construct almost-Bieberbach subgroups of B-4/Gamma(3)(P-4) and B-3/Gamma(4)(P-3), and we exhibit explicit elements of order 5 in B-5/Gamma(3)(P-5). (C) 2019 Elsevier Inc. All rights reserved.